A TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY

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1 U.P.B. Sci. Bull., Series A, Vol. 78, Iss. 2, 206 ISSN A TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY İbrahim Çaak I his paper we obai a Tauberia codiio i erms of he weighed classical corol modulo for he weighed mea mehod of summabiliy. Some addiioal resuls are also give. Keywords: summabiliy by he weighed mea, Tauberia codiios ad heorems, weighed classical ad geeral corol modulo. MSC200: 40A 0, 40E 05, 40G 05.. Iroducio Le p = ) be a sequece of oegaive real umbers wih p 0 > 0 ad := p k ). ) k=0 The -h weighed mea of u = u ) are defied by σ,pu) ) := p k u k. A sequece u ) is said o be summable by he weighed mea mehod N, p) o l, wrie as u l N, p), if lim σ),pu) = l. 2) Le X ad Y be wo sequece spaces ad A = a k ) be a ifiie marix. If for each x X he series A x) = k=0 a kx k coverges for each ad he sequece Ax = A x) Y we say ha he marix A maps X io Y. By X, Y ) we deoe he se of all marices which map X io Y. Le c be he se of all coverge sequeces. A marix A is called regular if A c, c) ad lim A x = lim k x k for all x c. The marix represeaio of weighed mea mehod N, p) is deoed by W = w k ), where w k is defied by w k = p k if k ad w k = 0 oherwise. I is kow ha N, p) summabiliy mehod is regular, i. e, W c, c) reg if ad oly if ) holds. k=0 Professor, Deparme of Mahemaics, Ege Uiversiy, Turkey, ibrahim.caak@ege.edu.r 9

2 92 İbrahim Çaak if The Silverma-Toepliz heorem saes ha A = a k ) is regular if ad oly R) A = su k a k <, R2) lim a k = 0 for each k, R3) lim k=0 a k =. If he limi lim u = l 3) exiss, he 2) is saisfied. However, he coverse is o always rue. Noice ha 2) implies 3) uder cerai codiios), which is called a Tauberia codiio. Ay heorem which saes ha covergece of sequeces follows from N, p) summabiliy mehod ad some Tauberia codiio is said o be a Tauberia heorem for N, p) summabiliy mehod. If = for all oegaive, he N, p) summabiliy mehod reduces o Cesàro summabiliy mehod. The backward differece of u ) is defied by u = u u for all oegaive, where u = 0. The differece bewee u ad is -h weighed mea σ,pu), ) which is called he weighed Kroecker ideiy [2] is give by he ideiy where u σ ),pu) = V 0),p u), 4) V 0),p u) := P k u k. The weighed classical corol modulo of u ) is deoed by ω,pu) 0) = u ad he weighed geeral corol modulo of ieger order m of u ) is defied i [2] by ω,p m) u) = ω,p m ) u) p k ω m ) P k,p u). k= k=0 For each ieger m 0, we defie σ,p m) u) by σ,p m) p k σ m ) u) = P k,p u), m k=0 u, m = 0 A sequece u ) is said o be slowly oscillaig [5] if lim m,x m x ) = 0. I erms of ϵ > 0 ad δ, his defiiio is equivale o he followig: for ay give ϵ > 0, here exis δ = δϵ) > 0 ad he posiive ieger N = Nϵ) such ha x m x < ϵ if Nϵ) ad m + δ). Our aim i his paper is o obai a Tauberia codiio i erms of he weighed classical corol modulo for N, p) summabiliy mehod. Some addiioal resuls are also give.

3 A Tauberia heorem for he weighed mea mehod of summabiliy The Resul We prove he followig Tauberia heorem for N, p) summabiliy mehod. Theorem 2.. Le ) be a sequece of oegaive umbers such ha p 0 > 0, lim if = O),, 5) P [λ] > for every λ >, 6) where [λ] deoes he iegral par of he produc λ, ad le u l N, p). The u ) coverges o l if for some > [λ] j=+ j,p u) = o), λ +. 7) Noe ha he codiio 6) imposed o he sequece ) was used i [4]. Remark 2.. We oe ha if j,p u) = log v, > 8) j= for some O-Regularly varyig sequece v ), he 8) is equivale o see [6]) j ω 0) j,p u) = O), >. j= if We remid he reader ha a posiive sequece u ) is O-Regularly varyig [] u [λ] u <, for λ >. 3. Lemmas We eed he followig Lemmas for he proof of Theorem 2.. Lemma 3.. [2]) Le u = u ) be a sequece of real umbers. For λ > ad sufficiely large, u σ ),pu) = P ) [λ] σ ) P [λ] P [λ],pu) σ),pu) where [λ] deoes he ieger par of λ. Lemma 3.2. [2]) For a sequece u ), σ ),pu) = V 0),p u). P [λ] k [λ] p k k=+ j=+ u j,

4 94 İbrahim Çaak P where, ad For a sequece u = u ), we defie ) ) P u = m m P P P u ) ) u = u 0 ) u = u. = P ) ) P u, m Lemma 3.3. [3]) For a sequece u ) ad ay ieger m, ) ω,p m) P u) = V,p m ) u). 9) m 4. Proof of Theorem 2. By Lemma 3., u σ,pu) ) By 6), we have Hece, we have P [λ] σ ) P [λ] P [λ],pu) σ),pu) + P [λ ] P [λ ] = P [λ] σ ) P [λ] P [λ],pu) σ),pu) lim if + Sice u ) is N, p) summable o l, boh he limis ad lim σ) [λ ],p u) = l lim σ),pu) = l exis. Therefore, we have, by ), P [λ] P [λ] σ ) P [λ] P [λ],pu) σ) [λ] j=+ u j. 0) <. ) P [λ] σ ) P [λ] P [λ],p u) l P [λ] σ ) P [λ] P,pu) l.,pu) = 0. 2)

5 A Tauberia heorem for he weighed mea mehod of summabiliy 95 For he secod erm o he righ-had side of 0) we obai [λ] j=+ u j = From 3), we have [λ] j=+ [λ] j=+ [λ] j=+ [λ] j=+ p j P j P j p j u j p j P j ω 0) j,p u) pj P j [λ] ) s [λ] ) s ) s s [λ] j=+ [λ] j=+ [λ] j=+ u j λ ) s From 2) ad 4), we have u σ,pu) ) λ ) s j,p u) ω 0) j,p u) j j,p u) [λ] j=+ [λ] j=+, where s + = j,p u) j,p u) 3). 4) = 0. 5) Leig λ + i 5) ad akig 7) io accou, we coclude ha u σ,pu) ) = 0. 6) This complees he proof of Theorem Some addiioal resuls If we replace he N, p) summabiliy of u ) i Theorem 2. by he summabiliy of σ,pu)) ) ad V,p 0) u)), we have he followig heorems. Theorem 5.. Le ) be a sequece of oegaive umbers such ha p 0 > 0, he codiios 5) ad 6) are saisfied ad le σ,pu) ) l N, p). The u l N, p) if for some > [λ] j=+ j V 0) j,p u) = o), λ +. 7)

6 96 İbrahim Çaak Proof. If we replace u = u ) by σu) = σ,pu)) ) i ω,pu) 0) = u, we obai ha ω,pσu)) 0) = σ,pu). ) By Lemma 3.2, σ ),pu) = V 0),p u). All he codiios of Theorem 2. are saisfied ad he codiio 7) becomes 7). This complees he proof of Theorem 5.. Theorem 5.2. Le ) be a sequece of oegaive umbers such ha p 0 > 0, he codiios 5) ad 6) are saisfied ad ad le V,p 0) u) l N, p). The u ) is slowly oscillaig if for some > [λ] j=+ j ω ) j,p u) = o), λ +. 8) Proof. If we replace u = u ) by V 0) u) = V,p 0) u)) i ω,pu) 0) = u, we obai ha ω,pv 0) 0) u)) = V,p 0) u). By Lemma 3.3, V 0),p u) = ω ),pu). All he codiios of Theorem 2. are saisfied ad he codiio 7) becomes 8). So, we have covergece of V,p 0) u)) o l. I follows from Lemma 3.2 ha σ,pu) ) σ m,pu) ) p k = V 0) P k,p u) k k=m+ for > m. By he codiio 5) ad he boudedess of V,p 0) u)), we have σ,pu) ) σ m,pu) ) C k C m ) k=m+ for some cosa C > 0. Takig he limi of boh sides of he las iequaliy as m, ad m, we obai ha σ),pu)) is slowly oscillaig. By Kroecker ideiy, u ) is slowly oscillaig. This complees he proof of Theorem Examples ad a applicaio for Theorem 2. If we ake = for all oegaive, he summabiliy by he weighed mea mehod N, p) reduces o he Cesàro summabiliy mehod. We have he followig examples of Theorem 2.. Example 6.. A Cesàro summable sequece u ) o l coverges o l i he ordiary sese if [λ] j=+ j, u) = o), λ +. 9)

7 A Tauberia heorem for he weighed mea mehod of summabiliy 97 If ω 0), u) = a for some bouded sequece a ) i Example 6., he we have he followig example. Example 6.2. A Cesàro summable sequece u ) o l coverges o l i he ordiary sese if where a ) is a bouded sequece. [λ] j=+ a j j = o), λ +, 20) We have he followig resul as a applicaio of Theorem 2.. A applicaio. Le ) be a sequece of oegaive umbers such ha p 0 > 0, he codiios 5) ad 6) are saisfied ad le u l N, p). The u ) coverges o l if j,p u) = log v 2) j= for some O-Regularly varyig sequece v ) ad for some >. Proof. Le he codiios 5) ad 6) be saisfied ad le u l N, p). If j= j,p u) = log v for some O-Regularly varyig sequece v ) ad for some >, he i is easy o show ha he codiio 7) is saisfied. Ideed, he lef side of he codiio 7) becomes log v [λ] log v ), which is o) as λ + by he defiiio of O-Regularly varyig sequece. R E F E R E N C E S [] V. G. Avakumović, Sur ue exesios de la codiio de covergece des heorems iverses de sommabilie, C. R. Acad. Sci. Paris, ), [2] İ. Çaak ad Ü. Tour, Some Tauberia heorems for he weighed mea mehods od summabiliy, Compu. Mah. Appl., 6220), [3] İ. Çaak ad Ü. Tour, Some geeral Tauberia heorems for he weighed mea summabiliy mehod, Compu. Mah. Appl., 63202), [4] F. Móricz ad B. E. Rhoades, Necessary ad sufficie Tauberia codiios for cerai weighed mea mehods of summabiliy II, Aca Mah. Hugar., ), No.4, [5] R. Schmid, Über divergee Folge ud lieare Mielbilduge, Mah. Z., 22925),

8 98 İbrahim Çaak [6] V. B. Saojevic, Fourier ad rigoomeric rasforms wih complex coefficies regularly varyig i mea, Fourier aalysis Oroo, ME, 992), , Lecure Noes i Pure ad Appl. Mah., 57, Dekker, New York, 994.